# Unified representation of the family of L-functions

## Abstract

The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.

AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

## 1 Introduction

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions $\mathbb{N}=\left\{1,2,\dots \right\}$, ${\mathbb{N}}_{0}=\left\{0,1,2,\dots \right\}=\mathbb{N}\cup \left\{0\right\}$, ${\mathbb{Z}}^{+}=\left\{1,2,3,\dots \right\}$, ${\mathbb{Z}}^{-}=\left\{-1,-2,\dots \right\}$. Also, as usual denotes the set of integers, denotes the set of real number and denotes the set of complex numbers. We assume that $ln\left(z\right)$ denotes the principal branch of the multi-valued function $ln\left(z\right)$ with the imaginary part $\mathrm{\Im }\left(ln\left(z\right)\right)$ constrained by $-\pi <\mathrm{\Im }\left(ln\left(z\right)\right)\le \pi$.

Recently, the first author  introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:

${g}_{a,b}\left(x;t,k,\beta \right):=\frac{{2}^{1-k}{t}^{k}{e}^{tx}}{{\beta }^{b}{e}^{t}-{a}^{b}}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\beta }\left(x;k,a,b\right)\frac{{t}^{n}}{n!},$
(1)

where ($|t|<2\pi$ when $\beta =a$; $|t|<|blog\left(\frac{\beta }{a}\right)|$ when $\beta \ne a$; $k\in {\mathbb{N}}_{0}$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\left\{0\right\}$).

For the special values of a, b, k, b and β, the polynomials ${\mathcal{Y}}_{n,\beta }\left(x;k,a,b\right)$ provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

Remark 1.1 If we set $k=a=b=1$ in (1), we get a special case of the generalized Bernoulli polynomials ${\mathcal{Y}}_{n,\beta }\left(x,k,1,1\right)$, that is, the so-called Apostol-Bernoulli polynomials ${\mathcal{B}}_{n}\left(x,\beta \right)$ generated by

$\frac{t}{\beta {e}^{t}-1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n}\left(x,\beta \right)\frac{{t}^{n}}{n!}$

(cf. ).

Remark 1.2 By substituting $k+1=-a=b=1$ in (1), we are led to Apostol-Euler polynomials ${\mathcal{E}}_{n}\left(x,\beta \right)$ which are defined by means of the following generating function:

$\frac{2}{\beta {e}^{t}+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n}\left(x,\beta \right)$

(cf. ).

Remark 1.3 Setting $k=-a=b=1$ into (1), we get the Apostol-Genocchi polynomials ${\mathcal{G}}_{n}\left(x,\beta \right)$ which are defined by means of the following generating function:

$\frac{2t}{\beta {e}^{t}+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n}\left(x,\beta \right)\frac{{t}^{n}}{n!}$

(cf. ).

In terms of a Dirichlet character χ of conductor $f\in \mathbb{N}$, Ozden et al.  extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [, p.2783])

Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers ${\mathcal{Y}}_{n,\chi ,\beta }\left(k,a,b\right)$ and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials ${\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)$ are given by the following generating functions:

${F}_{\chi ,\beta }\left(t;k,a,b\right)={2}^{1-k}{t}^{k}\sum _{j=1}^{f}\frac{\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{e}^{jt}}{{\beta }^{bf}{e}^{ft}-{a}^{bf}}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\chi ,\beta }\left(k,a,b\right)\frac{{t}^{n}}{n!},$
(2)

where ($|t|<2\pi$ when $\beta =a$; $|t|<|blog\left(\frac{\beta }{a}\right)|$ when $\beta \ne a$; $k\in {\mathbb{N}}_{0}$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\left\{0\right\}$) and

${\mathfrak{H}}_{\chi ,\beta }\left(x,t;k,a,b\right)={F}_{\chi ,\beta }\left(t,k;a,b\right){e}^{tx}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)\frac{{t}^{n}}{n!}$
(3)

($|t|<2\pi$ when $\beta =a$; $|t|<|blog\left(\frac{\beta }{a}\right)|$ when $\beta \ne a$; $k\in {\mathbb{N}}_{0}$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\left\{0\right\}$).

Remark 1.5 Substituting $k=a=b=\beta =1$ into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:

$\sum _{j=1}^{f}\frac{\chi \left(j\right)t{e}^{jt}}{{e}^{ft}-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,\chi }\frac{{t}^{n}}{n!}$
(4)

(cf. ).

## 2 Unification of the L-functions

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials ${\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)$ in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate ${\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)$ for negative integer values of n.

Throughout this section, we assume that $\beta \in \mathbb{C}$ with $|\beta |<1$ and $s\in \mathbb{C}$.

By substituting (1) into (2), we obtain the following functional equation:

${F}_{\chi ,\beta }\left(t;k,a,b\right)=\frac{1}{{f}^{k}}\sum _{j=1}^{f}\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{g}_{{a}^{f},b}\left(\frac{j}{f},tf;k,{\beta }^{f}\right).$
(5)

By using this functional equation, we arrive at the following theorem.

Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have

${\mathcal{Y}}_{n,\chi ,\beta }\left(k,a,b\right)={f}^{n-k}\sum _{j=1}^{f}\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{\mathcal{Y}}_{n,{\beta }^{f}}\left(\frac{j}{f};k,{a}^{f},b\right).$
(6)

By using (5), we modify (3) as follows:

${\mathfrak{H}}_{\chi ,\beta }\left(x,t;k,a,b\right)=\frac{1}{{f}^{k}}\sum _{j=1}^{f}\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{g}_{{a}^{f},b}\left(\frac{j+x}{f},tf;k,{\beta }^{f}\right).$
(7)

By using (7), we derive the following result.

Corollary 2.2 Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

${\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)={f}^{n-k}\sum _{j=1}^{f}\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{\mathcal{Y}}_{n,{\beta }^{f}}\left(\frac{j+x}{f};k,{a}^{f},b\right).$
(8)

By applying the Mellin transformation to the generating function (1), Ozden et al. [, p.2784 Equation (4.1)] gave an integral representation of the unified zeta function ${\zeta }_{\beta }\left(s,x;k,a,b\right)$:

${\zeta }_{\beta }\left(s,x;k,a,b\right)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_{0}^{\mathrm{\infty }}{t}^{s-k-1}{g}_{a,b}\left(x;-t;k,\beta \right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{1em}{0ex}}\left(min\left\{\mathrm{\Re }\left(s\right),\mathrm{\Re }\left(x\right)\right\}>0\right),$
(9)

where the additional constraint $\mathrm{\Re }\left(x\right)>0$ is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [, p.2784 Equation (4.1)] defined the unified zeta function ${\zeta }_{\beta }\left(s,x;k,a,b\right)$ as follows:

${\zeta }_{\beta }\left(s,x;k,a,b\right)={\left(-\frac{1}{2}\right)}^{k-1}\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^{bm}}{{a}^{b\left(m+1\right)}{\left(m+x\right)}^{s}}\phantom{\rule{1em}{0ex}}\left(\beta \in \mathbb{C}\left(|\beta |<1\right);s\in \mathbb{C}\left(\mathrm{\Re }\left(s\right)>1\right)\right).$
(10)

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$: (11)

in terms of the generating function ${\mathfrak{H}}_{\chi ,\beta }\left(x,t;k;a,b\right)$ defined in (7). By substituting (9) into ( 11), we obtain

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)=\frac{1}{{f}^{k+s}}\sum _{j=1}^{f}\chi \left(j\right){\left(\frac{\beta }{a}\right)}^{bj}{\zeta }_{{\beta }^{f}}\left(s,\frac{j+x}{f};k,{a}^{f},b\right)$
(12)

where ($\beta \in \mathbb{C}$ ($|\beta |<1$); $s\in \mathbb{C}$ ($\mathrm{\Re }\left(s\right)>1$)).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ as follows.

Definition 2.3 Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. For $s,\beta \in \mathbb{C}$ ($|\beta |<1$), we define a two-variable unified L-function ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ by

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)={f}^{-k}{\left(-\frac{1}{2}\right)}^{k-1}\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^{bm}\chi \left(m\right)}{{a}^{b\left(m+f\right)}{\left(m+x\right)}^{s}}\phantom{\rule{1em}{0ex}}\left(\beta \in \mathbb{C}\left(|\beta |<1\right);\mathrm{\Re }\left(s\right)>1\right).$
(13)

Remark 2.4 If we substitute $x=1$ into (13), we get the unified L-function

${L}_{\chi ,\beta }\left(s;k,a,b\right):={L}_{\chi ,\beta }\left(s,1;k,a,b\right)$

by

${L}_{\chi ,\beta }\left(s;k,a,b\right)={f}^{-k}{\left(-\frac{1}{2}\right)}^{k-1}\sum _{m=1}^{\mathrm{\infty }}\frac{{\beta }^{bm}\chi \left(m\right)}{{a}^{b\left(m+f\right)}{m}^{s}},$

where ($\mathrm{\Re }\left(s\right)>1$, $\beta \in \mathbb{C}$ ($|\beta |<1$)).

Remark 2.5 Upon substituting $k=a=b=1$ and $\beta =\frac{\xi }{u}$ into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:

${l}_{1}\left(\frac{u}{\xi },s,\chi \right)={L}_{\chi ,\frac{\xi }{u}}\left(s,x;1,1,1\right),$

where, for a positive integer r, ξ is the r th root of 1.

${l}_{1}\left(\frac{u}{\xi },s;\chi \right)=\sum _{m=0}^{\mathrm{\infty }}{\left(\frac{\xi }{u}\right)}^{m}\frac{\chi \left(m\right)}{{\left(m+x\right)}^{s}}$

(cf. ).

Remark 2.6 Substituting $x=1$ into (13), we get a unification of the L-functions

${L}_{\chi ,\beta }\left(s,1;k,a,b\right)={L}_{\chi ,\beta }\left(s;k,a,b\right).$

Substituting $\chi \equiv 1$ into (13), we get a unification ${\zeta }_{\beta }\left(s,x;k,a,b\right)$ of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function

$\zeta \left(s,x\right)={\zeta }_{1}\left(s,x;1,1,1\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\left(n+x\right)}^{s}}$

(cf. [27, 28]) and the Riemann zeta function

$\zeta \left(s\right)={\zeta }_{1}\left(s,1;1,1,1\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{n}^{s}}$

are obvious special cases of the unified zeta function ${\zeta }_{\beta }\left(s,x;k,a,b\right)$ (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function $\mathrm{\Phi }\left(z,s,a\right)$ was given by Ozden et al. :

${\zeta }_{\beta }\left(s,x;k,a,b\right):={\left(-\frac{1}{2}\right)}^{k-1}{a}^{-b}\mathrm{\Phi }\left(\frac{{\beta }^{b}}{{a}^{b}},s,x\right),$
(14)

where the Hurwitz-Lerch zeta function is defined by

$\mathrm{\Phi }\left(z,s,x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{z}^{n}}{{\left(n+x\right)}^{s}},$

which converges for ($x\in {\mathbb{C}\mathrm{╲}\mathbb{Z}}_{0}^{-}$, $s\in \mathbb{C}$ when $|z|<1$; $\mathrm{\Re }\left(s\right)>1$ when $|z|=1$), where as usual

${\mathbb{Z}}_{0}^{-}={\mathbb{Z}}^{-}\cup \left\{0\right\}$

(cf. [27, 28]).

A relationship between the functions ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ and ${\zeta }_{\beta }\left(s,x;k,a,b\right)$ is provided by the next theorem.

Theorem 2.7 Let $s\in \mathbb{C}$. Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)={f}^{-s-k}\sum _{j=1}^{f}{\left(\frac{\beta }{a}\right)}^{jb}\chi \left(j\right){\zeta }_{{\beta }^{f}}\left(s,\frac{j+x}{f};k,{a}^{f},b\right).$
(15)

Proof Substituting $m=nf+j$, $j=1,2,\dots ,f$, $n=0,\dots ,\mathrm{\infty }$ into (13), we obtain

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)={\left(-\frac{1}{2}\right)}^{k-1}{f}^{-s-k}\sum _{j=1}^{f}{\left(\frac{\beta }{a}\right)}^{jb}\chi \left(j\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{\beta }^{bnf}}{{a}^{bnf}{\left(n+\frac{j+x}{f}\right)}^{s}}.$

After some algebraic manipulations, we arrive at the desired result. □

Remark 2.8 Substituting $a=b=k=1$ into (13), we have

${L}_{\chi ,\beta }\left(s,x;1,1,1\right)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^{m}\chi \left(m\right)}{{\left(m+x\right)}^{s}}\phantom{\rule{1em}{0ex}}\left(\mathrm{\Re }\left(s\right)>1,\beta \in \mathbb{C}\left(|\beta |<1\right)\right)$

which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:

$\sum _{j=1}^{f}\frac{\chi \left(j\right)t{\beta }^{j}{e}^{t\left(j+x\right)}}{{\beta }^{f}{e}^{tf}-1}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,\chi }\left(x,\beta \right)\frac{{t}^{n}}{n!}.$

Let f be an odd integer. If we set $a=-1$ and $k=0$ into (13), then we have

${L}_{\chi ,\beta }\left(s,x;1,-1,1\right)=2\sum _{m=1}^{\mathrm{\infty }}{\left(-1\right)}^{m}\frac{\chi \left(m\right){\beta }^{m}}{{\left(m+x\right)}^{s}}\phantom{\rule{1em}{0ex}}\left(\mathrm{\Re }\left(s\right)>1,\beta \in \mathbb{C}\left(|\beta |<1\right)\right),$

which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:

$\sum _{j=1}^{f}\frac{2\chi \left(j\right){\beta }^{j}{e}^{t\left(j+x\right)}}{{\beta }^{f}{e}^{tf}+1}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n,\chi }\left(x,\beta \right)\frac{{t}^{n}}{n!}$

(cf. ).

By using (15) and (14), we arrive at the following result.

Corollary 2.9 Let $s\in \mathbb{C}$. Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)={\left(-\frac{1}{2}\right)}^{k-1}{a}^{-fb}{f}^{-s-k}\sum _{j=1}^{f}{\left(\frac{\beta }{a}\right)}^{jb}\chi \left(j\right)\mathrm{\Phi }\left(\frac{{\beta }^{fb}}{{a}^{fb}},s,\frac{j+x}{f}\right).$

Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have

${L}_{\chi ,\beta }\left(1-n,x;k,a,b\right)=\frac{{\left(-1\right)}^{k}}{f}\frac{\left(n-1\right)!}{\left(n+k-1\right)!}{\mathcal{Y}}_{n+k-1,\chi ,\beta }\left(x;k,a,b\right).$
(16)

Proof By substituting $s=1-n$ into (15), we get

${L}_{\chi ,\beta }\left(1-n,x;k,a,b\right)={f}^{n-1-k}\sum _{j=1}^{f}{\left(\frac{\beta }{a}\right)}^{jb}\chi \left(j\right){\zeta }_{{\beta }^{f}}\left(1-n,\frac{j+x}{f};k,{a}^{f},b\right).$

By using Theorem 7 in , we get By substituting (8) into the above, we arrive at the desired result. □

Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ defined by (13). We thus have (cf. )

$L\left(s,x;\chi \right)=\sum _{m=0}^{\mathrm{\infty }}\frac{\chi \left(m\right)}{{\left(m+x\right)}^{s}}$

and

$L\left(s;\chi \right)=\sum _{m=1}^{\mathrm{\infty }}\frac{\chi \left(m\right)}{{m}^{s}},$

where $\mathrm{\Re }\left(s\right)>1$. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have

$L\left(1-n;\chi \right)=-\frac{{B}_{n,\chi }}{n},$

where $n\in {\mathbb{Z}}^{+}$ and ${B}_{n,\chi }$, the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that $L\left(s;\chi \right)$ is non-zero at $s=1$. Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at $s=1$ (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

## 3 Applications

In this section, by using (16) and the following formula, which was proved by Ozden et al. [, Theorem 5, Equation (3.10)]

${\mathcal{Y}}_{n,\chi ,\beta }\left(x;k,a,b\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){x}^{n-j}{\mathcal{Y}}_{j,\chi ,\beta }\left(k,a,b\right),$
(17)

we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),

${L}_{\chi ,\beta }\left(1-n,x;k,a,b\right)=\frac{{x}^{n+k-1}}{f{\prod }_{l=0}^{k-1}\left(n+l\right)}\sum _{j=0}^{n+k-1}\left(\genfrac{}{}{0}{}{n+k-1}{j}\right)\frac{1}{{x}^{j}}{\mathcal{Y}}_{j+k-1,\chi ,\beta }\left(k,a,b\right).$

From the above equation, we arrive at the following theorem.

Theorem 3.1 Let $x\ne 0$. Let χ be a Dirichlet character of conductor f. Then we have

${L}_{\chi ,\beta }\left(s,x;k,a,b\right)=\frac{{x}^{k-s}}{f{\prod }_{l=0}^{k-1}\left(s-1-l\right)}\sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{k-s}{j}\right)\frac{1}{{x}^{j}}{\mathcal{Y}}_{j+k-1,\chi ,\beta }\left(k,a,b\right).$

The function ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ is an analytic function at $s=0$. We now compute the value of this function at this point as follows:

${L}_{\chi ,\beta }\left(0,x;k,a,b\right)=\frac{{x}^{k}}{{\left(-1\right)}^{k}f{\prod }_{l=0}^{k-1}\left(1+l\right)}\sum _{j=0}^{k}\left(\genfrac{}{}{0}{}{k}{j}\right)\frac{1}{{x}^{j}}{\mathcal{Y}}_{j+k-1,\chi ,\beta }\left(k,a,b\right).$

The function ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ is a meromorphic function. This function has simple poles which are

$s=1,2,3,\dots ,k.$

The residues of this function at the simple poles at $s=1$ and $s=k$ are given, respectively, as follows:

${Res}_{s=1}\left\{{L}_{\chi ,\beta }\left(s,x;k,a,b\right)\right\}=\frac{{x}^{k-1}}{f{\left(-1\right)}^{k}{\prod }_{l=0}^{k-1}\left(2+l\right)}\sum _{j=0}^{k-1}\left(\genfrac{}{}{0}{}{k-1}{j}\right)\frac{1}{{x}^{j}}{\mathcal{Y}}_{j+k-1,\chi ,\beta }\left(k,a,b\right)$

and

${Res}_{s=k}\left\{{L}_{\chi ,\beta }\left(s,x;k,a,b\right)\right\}=\frac{{\mathcal{Y}}_{k-1,\chi ,\beta }\left(k,a,b\right)}{f{\prod }_{l=0}^{k-2}\left(k-1-l\right)}.$

Remark 3.2 Simsek (cf. [20, 21]) defined a twisted two-variable L-function ${L}_{\xi ,q}^{\left(h\right)}\left(s,x;\chi \right)$ as follows:

${L}_{\xi ,q}^{\left(h\right)}\left(s,x;\chi \right)=\sum _{m=0}^{\mathrm{\infty }}\frac{\chi \left(m\right){\varphi }_{\xi }\left(m\right){q}^{hm}}{{\left(x+m\right)}^{s}}-\frac{log{q}^{h}}{s-1}\sum _{m=0}^{\mathrm{\infty }}\frac{\chi \left(m\right){\varphi }_{\xi }\left(m\right){q}^{hm}}{{\left(x+m\right)}^{s-1}},$

where $q\in \mathbb{C}$ ($|q|<1$); ${\xi }^{r}=1$ ($r\in \mathbb{Z}$); $\xi \ne 1$. Observe that if $\xi =1$, then ${L}_{\xi ,q}^{\left(h\right)}\left(s,x;\chi \right)$ is reduced to the work of Kim .

Relationship between the function ${L}_{\chi ,\beta }\left(s,x;k,a,b\right)$ and ${L}_{\xi ,q}^{\left(h\right)}\left(s,x;\chi \right)$ is given as the following result.

Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have

${L}_{1,\frac{{\beta }^{b}}{{a}^{b}}}^{\left(b\right)}\left(s,x;\chi \right)={\left(-2\right)}^{k}{a}^{bf}{f}^{k}\left({L}_{\chi ,\beta }\left(s,x;k,a,b\right)-\frac{log{q}^{h}}{s-1}{L}_{\chi ,\beta }\left(s-1,x;k,a,b\right)\right).$

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function $\mathrm{\Phi }\left(z,s,a\right)$, Hurwitz zeta function $\zeta \left(s,x\right)$ and L-function (cf. ); see also the references cited in each of these earlier works.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

Both authors are partially supported by Research Project Offices Akdeniz Universities and the Commission of Scientific Research Projects of Uludag University Project number UAP(F) 2011/38 and 2012/16. We would like to thank referees for their valuable comments.

## Author information

Authors

### Corresponding author

Correspondence to Hacer Ozden.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors completed the paper together. Both authors read and approved the final manuscript.

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Ozden, H., Simsek, Y. Unified representation of the family of L-functions. J Inequal Appl 2013, 64 (2013). https://doi.org/10.1186/1029-242X-2013-64

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• DOI: https://doi.org/10.1186/1029-242X-2013-64

### Keywords

• Bernoulli numbers
• Bernoulli polynomials
• Euler numbers
• Euler polynomials
• Genocchi numbers
• Genocchi polynomials
• Dirichlet L-functions
• Hurwitz zeta function
• Riemann zeta function 